representations. Implicit in our treatment is the need to think in terms of representations that have a tensor structure: we note such structures where they appear; but these structures play a much more important role than just their manifestation in the handling of angular momentum and spin. We continue with the details of the coupling of spins and angular momenta. The techniques are fundamental to handling finite many-body quantum systems. The Clebsch–Gordan coefficients that are encountered here are an essential topic that must be mastered by a researcher in nuclear structure, whether theorist or experimentalist. We give a brief introduction to ‘recoupling’ coefficients, 6-j and 9-j symbols. The manipulation of recoupling is straightforward but takes practice to achieve mastery: that is left for our more advanced reader to take up at a later stage in the series. We complete the angular momentum topics with details of vector and tensor operators. This is also a topic that must be mastered by the nuclear structure researcher, such that they have a clear appreciation of the power of the Wigner–Eckart theorem.
The focus then turns to identical particles and the representation of many-body states and operators. A heavy emphasis is placed on second quantization or the occupation number representation. We pay particular attention to how this language can be used to formulate solvable models of systems with correlations. Such systems exhibit properties that lie completely outside of any classical mechanical concepts. The quasi-spin formalism is a powerful language for the description of many-fermion systems that form Cooper pairs and is developed in detail. We also introduce the Lipkin model: this is a ‘toy’ model, i.e. it is not realised in nature (it is too simple). But it is exactly solvable and so can be used to test many-body approximation methods. Many-body approximations such as Hartree–Fock theory will be handled later in the series.
We then turn to the role of group theory and of algebraic structures in quantum mechanics. These two topics are closely related in quantum mechanics because of the close relationship between Lie groups and Lie algebras. We give a basic introduction to these topics, using what has been learned via angular momentum theory. We particularly emphasize the role that groups and algebras play in quantum mechanics, both as a way to a deeper understanding of the subject and as a set of tools for formulating models. We provide an introduction to Young diagrams and their manipulation. We take a few steps into the Cartan theory of Lie algebras, sufficient to acquire a deeper appreciation for the mathematics behind their application to quantum systems, especially ladder operators and spectrum generating algebras.
We complete the volume with standard treatments of perturbation theory, the variational method, and a brief handling of the quantization of the electromagnetic field and its interactions with matter.
We have aimed to focus on the quantum mechanics needed for taking up research into the nuclear many-body problem, without going into the details of nuclear modelling and approximation methods. These require familiarity with nuclear data and transformation processes, which adds another ‘dimension’ to the path to mastery of the subject. These steps will be taken later in the series.
Author biographies
Kris Heyde
Kris Heyde is Professor Emeritus in the Department of Physics and Astronomy at the University of Gent. He continues to work on joint research projects with both experimental and theoretical nuclear physicists. His research interests are on one side directed to learn how collectivity emerges starting from a microscopic point of view. He has a long-standing deep interest in the presence of shape coexistence in atomic nuclei, in particular studying the conditions needed to be realised throughout the nuclear mass table.
John L Wood
John L Wood is a Professor Emeritus in the School of Physics at Georgia Institute of Technology. He continues to collaborate on research projects in both experimental and theoretical nuclear physics. Special research interests include nuclear shapes and systematics of nuclear structure.
IOP Publishing
Quantum Mechanics for Nuclear Structure, Volume 2
An intermediate level view
Kris Heyde and John L Wood
Chapter 1
Representation of rotations, angular momentum and spin
The various representations of rotations in physical space, (3,R) and Hilbert space (n,C) are developed in detail. This leads to an in-depth treatment of the representation of states of well-defined spin and angular momentum in quantum systems. The peculiarities of the physics of spin-12 systems (spinors) are outlined. The tensorial character of representations is implicit in the treatment. The Schwinger and Bargmann representations are introduced in some detail; and this leads to SU(2) coherent states (which are important for more advanced group representation theory).
Concepts: Euler angles; matrix representations; Pauli spin matrices; ket rotations; SU(2) and SO(3) tensor representations; Schwinger representation; spherical harmonics as Cartesian tensors; spin-12 neutron interferometry; Bargmann space; measure of a space; SU(2) coherent states; non-unitary representations.
Angular momentum and spin are dynamical variables that are fundamental to finite systems in quantum mechanics, i.e. for molecules, atoms, nuclei and hadrons. To fully handle the quantum mechanics of these systems, the mathematical representation of rotations is fundamental. Some elements of these issues in quantum mechanics are introduced in Volume 1. Namely, the concept of a group, the use of matrices, the distinction between rotations in physical space, (3, R), and Hilbert space is presented in chapter 10; and the basic quantization of spin and angular momentum, using algebraic methods, is presented in chapter 11. Further, the facility with which these methods reduce the solution of central force problems in quantum mechanics to simple algebraic problems in terms of a single (radial) degree of freedom is presented in chapter 12.
The mathematical representation of rotations is a rich paradigm for the whole of quantum mechanics. In this chapter, a wide range of mathematical tools is introduced. Matrix algebra and the algebra of polynomials in real and complex variables feature prominently. The peculiar physics of spin-12 particles and spinors is presented. But, the primary aim is to initiate a language that is suitable for the theoretical formulation of finite many-body quantum systems. Group theory and Lie algebras are implicit in the material presented in this chapter: the groups SO(2), SO(3) and SU(2) feature prominently in their behind-the-scene role. The road into many-body systems necessitates more complicated groups such as SU(3): some of the material in this chapter is intended to ‘pave’ this road.
1.1 Rotations in (3, R)
One way to describe a rotation in (3, R) is in terms of rotation in a plane through a specified angle1. This is defined in terms of an axis of rotation nˆ and an angle ϕ. The axis nˆ is perpendicular to the plane defined by the initial and final orientations of the vectors V⃗, V⃗′: R(ϕ)V⃗=V⃗′. The difficulty lies in ascertaining the direction of nˆ. Although this ‘axis-angle’ parameterisation or Darboux parameterisation is simple in principle, it is difficult to use in practice.
The most widely used practical parameterisation of rotations in (3, R) is in terms of Euler rotations. Consider a space-fixed coordinate frame Oxyz and a body-fixed coordinate frame Ox¯y¯z¯. The orientation of an object can be specified by the rotation R that rotates the Ox¯y¯z¯ frame into the Oxyz frame. This can be done in three steps as illustrated in figure 1.1.
Figure
